Manipulation and/or analysis of hierarchical data

ABSTRACT

Embodiments of methods, apparatuses, devices and/or systems for manipulating hierarchical sets of data are disclosed. In particular, methods, apparatus devices and or/or systems for analyzing hierarchical data are disclosed. In a particular example, rooted partial subtrees of partial subtrees of a tree are enumerated and associated with target numerals according to an association of trees and numerals. A probe numeral, representing information of interest, may be compared with target numerals to find a match. In one particular example, such a match indicates a presence of particular information in the tree.

RELATED APPLICATION

This application claims priority to U.S. Provisional Patent ApplicationSer. No. 60/675,987 filed Apr. 29, 2005, entitled MANIPULATION AND/ORANALYSIS OF HIERARCHICAL DATA, assigned to the assignee of claimedsubject matter.

BACKGROUND

This disclosure is related to hierarchical data arrangements and, moreparticularly, to manipulating such data arrangements.

In a variety of fields, data or a set of data, may be represented in ahierarchical fashion. This form of representation may, for example,convey information, such as particular relationships between particularpieces of data and the like. However, manipulating such datarepresentations is not straight-forward, particularly where the data isarranged in a complex hierarchy. Without loss of generality, one examplemay include a relational database. Techniques for performing operationson such a database, for example, are computationally complex orotherwise cumbersome. A continuing need, therefore, exists foradditional techniques for manipulating data hierarchies.

BRIEF DESCRIPTION OF THE DRAWINGS

Subject matter is particularly pointed out and distinctly claimed in theconcluding portion of the specification. Claimed subject matter,however, both as to organization and method of operation, together withobjects, features, and advantages thereof, may best be understood byreference of the following detailed description when read with theaccompanying drawings in which:

FIG. 1 is a schematic diagram of one embodiment of a tree;

FIG. 2 is a schematic diagram illustrating one embodiment of a binaryedge labeled tree (BELT);

FIG. 3 is a schematic diagram illustrating another embodiment of a BELT;

FIG. 4 is a table illustrating a particular embodiment of an associationbetween natural numerals and BELTs;

FIG. 5 is a table illustrating a portion of the Kleene enumeration ofnon-composite numerals;

FIG. 6 is a schematic diagram of an embodiment of a node labeled tree;

FIG. 7 is a schematic diagram illustrating another embodiment of a nodelabeled tree;

FIG. 8 is a flow diagram illustrating a process embodiment to detectand/or locate information in hierarchical data of a target treeaccording to an embodiment;

FIG. 9 is a flow diagram illustrating a process embodiment to representa target tree as one or more numerals according to an embodiment of theprocess illustrated in FIG. 8;

FIG. 10A is a schematic diagram of a target tree according to anembodiment;

FIG. 10B is a schematic diagram of an embodiment of smaller trees of atarget tree according to an embodiment of the target tree shown in FIG.10A.

FIG. 11 is a schematic diagram of probe trees according to anembodiment;

FIGS. 12 through 15 are schematic diagrams illustrating, by way ofexample, rooted partial subtrees (RPSTs) according to an embodiment.

FIG. 16 is a flow diagram illustrating a process to enumerate RPSTs froma tree according to an embodiment.

FIG. 17 is a schematic diagram illustrating a representation of anembodiment of a subtree according to an embodiment.

FIGS. 18 and 19 are schematic diagrams illustrating applications of oneembodiment of a push operation according to an embodiment.

FIGS. 20, 21 and 22 are flow diagrams illustrating a process toenumerate RPSTs from a tree according to an embodiment.

FIGS. 23 through 31 are schematic diagrams illustrating, by way ofexample, a process to enumerate RPSTs from a tree according to anembodiment.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are setforth to provide a thorough understanding of claimed subject matter.However, it will be understood by those skilled in the art that claimedsubject matter may be practiced without these specific details. In otherinstances, well-known methods, procedures, components and/or circuitshave not been described in detail so as not to obscure claimed subjectmatter.

Some portions of the detailed description which follow are presented interms of algorithms and/or symbolic representations of operations ondata bits or binary digital signals stored within a computing systemmemory, such as a computer memory. These algorithmic descriptions and/orrepresentations are the techniques used by those of ordinary skill inthe data processing arts to convey the substance of their work to othersskilled in the art. An algorithm is here, and generally, considered tobe a self-consistent sequence of operations and/or similar processingleading to a desired result. The operations and/or processing involvephysical manipulations of physical quantities. Typically, although notnecessarily, these quantities may take the form of electrical and/ormagnetic signals capable of being stored, transferred, combined,compared and/or otherwise manipulated. It has proven convenient attimes, principally for reasons of common usage, to refer to thesesignals as bits, data, values, elements, symbols, characters, terms,numbers, numerals and/or the like. It should be understood, however,that all of these and similar terms are to be associated with theappropriate physical quantities and are merely convenient labels. Unlessspecifically stated otherwise, as apparent from the followingdiscussion, it is appreciated that throughout this specificationdiscussions utilizing terms such as “processing”, “computing”,“calculating”, “transforming,” “converting,” “factoring,” enumerating,”“representing,” “storing,” “associating,” “substituting,” “determining”and/or the like refer to the actions and/or processes of a computingplatform, such as a computer or a similar electronic computing device,that manipulates and/or transforms data represented as physicalelectronic and/or magnetic quantities and/or other physical quantitieswithin the computing platform's processors, memories, registers, and/orother information storage, transmission, and/or display devices.Further, unless specifically stated otherwise, processes describedherein, with reference to flow diagrams or otherwise, may also beexecuted and/or controlled, in whole or in part, by such a computingplatform.

In a variety of fields, data or sets of data may be represented in ahierarchical fashion. This form of representation may, for example,convey information, such as particular relationships between particularpieces of data and the like. However, manipulating such datarepresentations is not straight forward, particularly where the data isarranged in a complex hierarchy. Without loss of generality, one examplemay include a relational data base. Techniques for performing operationson such a data base for example, may be computationally complex orotherwise cumbersome. A continuing need, therefore, exists foradditional techniques for manipulating data hierarchies.

As previously discussed, in a variety of fields, it is convenient ordesirable to represent data, a set of data and/or other information in ahierarchical fashion. In this context, such a hierarchy of data shall bereferred to as a “tree.” In a particular embodiment, a tree may comprisea finite, rooted, connected, unordered, acyclic graph. This isillustrated here, for example, in FIG. 1 by embodiment 100. Asillustrated, the root of this particular embodiment encompasses node105. In addition to 105, there are eight other nodes designated 110 to140, respectively. Likewise, the nodes are connected by branchesreferred to, in this context, as edges. Thus, the nodes of this tree areconnected by eight edges. This embodiment, therefore, illustrates afinite tree that is rooted by node 105. Furthermore, the nodes areconnected, meaning, in this context, that a path exists between any twonodes of the tree. The tree is likewise acyclic, meaning here, that nopath in the tree forms a complete backtracking loop. Here, unorderedrefers to the notion that there is no implied ordering or precedenceamong nodes attached to a common node, despite the appearance ofordering in a graphical illustration.

As previously suggested, in a variety of contexts, it may be convenientand/or desirable to represent a hierarchy of data and/or otherinformation using a structure, such as the embodiment illustrated inFIG. 1. One particular embodiment, without loss of generality, of a treemay include edges that are labeled with data and/or other values.Likewise, in one particular embodiment, such data or values may belimited to binary data, that is, in this example, either a binary one ora binary zero. Here, such an embodiment may be referred to as a binaryedge labeled tree (BELT), as shall be discussed in more detailhereinafter.

One example of a BELT is illustrated by embodiment 200 of FIG. 2. Thus,as illustrated, the edges of the BELT shown in FIG. 2 are labeled witheither a binary zero or binary one. FIG. 3 illustrates anotherembodiment 300 of a different binary edge labeled tree. It is noted thatthis tree is similar to the embodiment of FIG. 2. Without belaboring thepresent discussion, additional descriptions of how BELTs may represent ahierarchy of data may be found in U.S. patent application Ser. No.11/005,859, filed on Dec. 6, 2004, by J. J. LeTourneau, titled,“Manipulating Sets of Hierarchical Data,” assigned to the assignee ofthe subject matter claimed herein.

The terms “non-composite numeral” and “primary numeral” shall be usedinterchangeably herein. Being trees, BELTs may also be enumerated. Thus,for this particular embodiment, although claimed subject matter is notlimited in scope in this respect, a method of enumerating a set of treesbegins with enumeration of an empty binary edge labeled tree and a onenode binary edge labeled tree. Here, the empty tree is associated withthe zero and has a symbolic representation as illustrated in FIG. 4(circle). Likewise, the one node tree, which holds no data, isassociated with the numeral one and has a graphical representation of asingle node. For higher positive natural numerals, however, thisembodiment of a method of enumerating a set of trees comprisespositioning a tree at location k, k being a positive numeral greaterthan three, where k is the product of u and v, u and v comprisingpositive numerals greater than one, such that the tree is formed by aunion of the trees at positions u and v. Likewise, for those locationsthat are not a product of other natural positive numerals greater thanone, that is, for locations that comprise non-composite numerals,denoted here by j, for example, j being a positive natural numeralgreater than one, a tree is positioned at location j such that the treeis formed by finding the first tree in the prior enumeration such thatthe binary edge labeled tree obtainable from this first tree byattaching a node to the particular tree as a new root node and labelingthe edge between the new root node and the prior root node with a binary“0” label is not in the enumeration at some position lower than j;however, if the binary edge labeled tree obtainable from that firsttree, as just described, is present in the enumeration with a binary “0”label for the new edge, but not with a binary “1” label, then the treeat position j is that tree with a binary “1” label for the new edge.This may be illustrated; for example in FIG. 4, as described in moredetail below.

However, for this particular embodiment, although claimed subject matteris not limited in scope in this respect, a method of enumerating a setof ordered trees may begin with enumeration of an empty binary edgelabeled tree and a one node binary edge labeled tree. Thus, the emptytree is associated with numeral zero and has a symbolic representationas illustrated in FIG. 4 (circle). Likewise, the one node tree, whichholds no data, is associated with numeral one and has a graphicalrepresentation of a single node. For higher positive natural numerals,ordered trees may be generated by a process described, for example, in“The Lexicographic Generation of Ordered Trees,” by S. Zaks, The Journalof Theoretical Computer Science, Vol. 10(1), pp. 63-82, 1980, orEnumerating Ordered Trees Lexicographically,” by M. C. Er, ComputationJournal, Vol. 28, Issue 5, pp. 538-542, 1985.

As illustrated, for this particular embodiment, and as previouslydescribed, the empty tree has zero nodes and is associated with numeralzero. Likewise, the one node tree root comprises a single node and isassociated with numeral one. Thus, to obtain the tree at position two, aroot node is attached and connected to the prior root node by an edge.Likewise, here, by convention, the edge is labeled with a binary zero.If, however, the tree formed by the immediately proceeding approach werepresent in the prior enumeration of trees, then a similar processembodiment is followed, but, instead, the new edge is labeled with abinary one rather than a binary zero. Thus, for example, in order toobtain the binary edge labeled tree for position three, a new root nodeis connected to the root node by an edge and that edge is labeled with abinary one.

Continuing with this example, to obtain the binary edge labeled tree forposition four, observe that numeral four is the product of numeral twotimes numeral two. Thus, a union is formed at the root of two trees,where, here, each of those trees is associated with the numeral two.Likewise, to obtain the binary edge labeled tree for position five,begin with the binary edge labeled tree for position two and follow thepreviously articulated approach of adding a root and an edge andlabeling it with a binary zero.

In this context, adding a root node and an edge and labeling it binaryzero is referred to as a “zero-push” operation and adding a root nodeand an edge and labeling it binary one is referred to as a “one-push”operation. Based at least in part on the prior description, for thisparticular embodiment, it may now be demonstrated that if k is anypositive natural numeral and a tree x is positioned at location k, thena non-composite numeral is associated with the zero-push of that treeand a non-composite numeral is associated with the one-push for thattree. Furthermore, the non-composite index of the zero-push of the treecomprises 2k−1, whereas the non-composite index of the one-push of thetree comprises 2k, where the index corresponds to the argument of thewell-known Kleene enumeration on positive natural numerals ofnon-composite numerals, as illustrated, for example, in part in FIG. 5.Thus, referring again to FIG. 4, the one-push of the root tree is thetree at position three. This follows from FIG. 5 since P(2*1)=P(2)=3.Likewise, the tree at position five is the zero-push of the tree atposition 2. Again, this follows from FIG. 5 since P(2*2−1)=P(3)=5.

In this context, the approach just described may be referred to asvectorizing non-composite numerals. In the embodiment just described,this was accomplished in pairs, although, of course, claimed subjectmatter is not limited in scope in this respect. This may be accomplishedin any number of numeral combinations, such as triplets, quadruplets,etc. Thus, using a quadruplet example, it is possible to construct treessuch that if k is any positive natural numeral and a tree x ispositioned at location k, then a non-composite numeral is associatedwith the zero-push of that tree, a non-composite numeral is associatedwith the one-push for that tree, a non-composite numeral is associatedwith the two-push for that tree, and a non-composite number isassociated with the three-push for that tree. Furthermore, the index ofthe non-composite numeral is such that for a zero-push of the tree, theindex comprises (4k−3), for a one-push of a tree, the index comprises(4k−2), for a two-push of a tree, the index comprises (4k−1), and for athree-push of a tree the index comprise (4k), where the indexcorresponds to the Kleene enumeration of non-composite numerals,P(index), such as provided in FIG. 5.

In the previously described enumeration of binary edged labeled trees, amechanism may be employed to reduce or convert and/or transform complexmanipulations of hierarchical data to multiplication of naturalnumerals. For example, if it is desired to combine, or merge at theirroots, two trees of hierarchical data, a complex task bothcomputationally and graphically, instead, for this particularembodiment, the two trees may be transformed to numerical data by usingthe previously described association embodiment between binary edgelabeled trees and natural numerals. The resulting numerical data fromthe prior conversion may then be multiplied, and the resulting productmay then be transformed to a binary edge labeled tree by using a tablelook up of the previously described association embodiment. It is notedthat a subtle distinction may be made between an enumeration embodimentand an association embodiment. Enumeration may comprise listing, in thisexample, a particular ordered embodiment of BELTs, whereas anassociation provides a relationship between, in this example, aparticular ordered embodiment of BELTs and natural numerals. It is, ofcourse, appreciated that many different enumeration and associationembodiments may be employed to execute the operations discussed aboveand hereinafter, and claimed subject matter is intended to cover allsuch enumeration and association embodiments.

Likewise, a process embodiment that is a reversal to the previouslydescribed embodiments may also be employed. Thus, complex hierarchies ofdata may be split or divided, when this is desired. For example, abinary edge labeled tree to be divided may be transformed to a piece ofnumerical data, such as by using the previously described associationembodiment. This data may then be factored into two pieces of numericaldata whose product produces the previously mentioned piece of numericaldata. These two pieces of numerical data may then be transformed totrees, again, by using the prior association embodiment, for example.

Another form of manipulating hierarchical sets of data may involveordering or hashing. This may be desirable for any one of a number ofdifferent operations to be performed on the sets of data. One approachis similar to the previously described embodiment. For example, it maybe desired to order a given set of trees. Doing so may involvetransforming the trees to numerical data, as previously described, usingan association embodiment. The numerical data may then be ordered andthe numerical data may then be transformed back to binary edge labeledtrees using the previously described association embodiment, or analternate association embodiment, for example.

It is noted that there may be any one of a number of different ways oftransforming from numerals or numerical data values to a binary edgelabeled tree or from a binary string to a binary edge labeled tree, andvice-versa. Nonetheless, a convenient method for doing so with thisparticular embodiment includes storing an array providing an associationembodiment between natural numerals, binary strings and binary edgelabeled trees, such as the embodiment previously described. Thus, onceit is desired to convert from one to the other, such as from a binarystring to a BELT, from a natural numeral to a BELT, or vice-versa, forexample, a table look up operation may be performed using theassociation embodiment.

Techniques for performing table look ups are well-known andwell-understood. Thus, this will not be discussed in detail here.However, it shall be appreciated that any and all of the previouslydescribed and/or later described processing, operations, conversions,transformations, manipulations, etc. of strings, trees, numerals, data,etc. may be performed on one or more computing platforms or similarcomputing devices, such as those that may include a memory to store anarray as just described, although, claimed subject matter is notnecessarily limited in scope to this particular approach. Thus, forexample, a hierarchy of data may be formed by combining two or morehierarchies of data, such as by applying a previously describedembodiment. Likewise, multiple hierarchies of data may be formed bysplitting or dividing a particular hierarchy of data, again, such as byapplying a previously described embodiment. Likewise, additionaloperations and/or manipulations of data hierarchies may be performed,such as ordering hierarchies of data and more. It is intended thatclaimed subject matter cover such embodiments.

Much of the prior discussion was provided in the context of binary edgelabeled trees. Nonetheless, as alluded to previously, binary edgelabeled trees and binary node labeled trees may be employed nearlyinterchangeably to represent substantially the same hierarchy of data.In particular, a binary node labeled tree may be associated with abinary edge labeled tree where the nodes of the binary node labeled treetake the same values as the edges of the binary edge labeled tree,except that the root node of the binary node labeled tree may comprise anode having a zero value or a null value. Thus, rather than employingbinary edge labeled trees, the previously described embodiments mayalternatively be performed using binary node labeled trees. As oneexample embodiment, operations and/or manipulations may be employedusing binary edge labeled trees and then the resulting binary edgelabeled tree may be converted to a binary node labeled tree. However, inanother embodiment, operations and/or manipulations may be performeddirectly using binary node labeled trees where a different associationembodiment, that is, in this example, one that employs binary nodelabeled trees, is employed.

In accordance with claimed subject matter, therefore, any tree,regardless of whether it is binary edge labeled, binary node labeled,non-binary, a feature tree, or otherwise, may be manipulated and/oroperated upon in a manner similar to the approach of the previouslydescribed embodiments. Typically, different association embodimentsshall be employed, depending at least in part, for example, upon theparticular type of tree. For example, and as shall be described in moredetail below in connection with FIG. 6, a node labeled tree in which thenodes are labeled with natural numerals or data values may be convertedto a binary edge labeled tree. Furthermore, this may be accomplishedwith approximately the same amount of storage. For example, for thisparticular embodiment, this may involve substantially the same amount ofnode and/or edge data label values.

As previously noted, claimed subject matter is not limited in scope tothis particular example, however, as illustrated in more detailhereinafter, the tree illustrated in FIG. 6 is converted to a binaryedge labeled tree through a sequence of processing depicted here asgraph operations, although such a conversion may alternatively beimplemented by operations implemented otherwise, one such example beinga computing platform, for example. Alternatively, it may be desirable,depending upon the particular embodiment, to convert trees to, forexample binary node labeled trees. Likewise, other embodiments in whichtrees of one form are converted to trees of another form are alsoincluded within the scope of the claimed subject. However, for thisparticular embodiment, it will be assumed that the association betweentrees and numerals, such as previously described, is depicted orenumerated in terms of binary edge labeled trees, as previouslyillustrated, for example. Thus, in this example, a particular tree,embodiment 600, is illustrated in FIG. 6, comprises a node labeled treerather than an edge labeled tree. Without belaboring the presentdiscussion, a process of converting a node labeled tree such as thatillustrated in FIG. 6 to a BELT may be found in U.S. patent applicationSer. No. 11/005,859, filed on Dec. 6, 2004, by J. J. LeTourneau, titled,“Manipulating Sets of Hierarchical Data,” assigned to the assignee ofthe presently claimed subject matter.

In another embodiment, however, a particular tree may include null typesor, more particularly, some node values denoted by the empty set. Thisis illustrated, for example, by the tree in FIG. 7, although, of course,this is simply one example. An advantage of employing null typesincludes the ability to address a broader array of hierarchical datasets. For example, without loss of generality and not intending to limitthe scope of claimed subject matter in any way, a null type permitsrepresenting in a relational database, as one example, situations wherea particular attribute does not exist. As may be appreciated, this isdifferent from a situation, for example, where a particular attributemay take on a numeral value of zero. Thus, it may be desirable to beable to address both situations when representing, operating upon and/ormanipulating hierarchical sets of data. A tree with nulls may beconverted to a tree without nulls as described in U.S. patentapplication Ser. No. 11/005,859, filed on Dec. 6, 2004, by J. J.LeTourneau, titled, “Manipulating Sets of Hierarchical Data,” assignedto the assignee of the presently claimed subject matter.

Likewise, in an alternative embodiment, a node labeled tree may comprisefixed length tuples of numerals. For such an embodiment, such multiplenumerals may be combined into a single numeral, such as by employingCantor pairing operations, for example. See, for example, Logical NumberTheory, An Introduction, by Craig Smorynski, pp, 14-23, available fromSpringer-Verlag, 1991. This approach should produce a tree to which thepreviously described embodiments may then be applied. Furthermore, forone embodiment, a tree in which nodes are labeled with numerals ornumerical data, rather than binary data, may be transformed to a binaryedge labeled tree and/or binary node labeled tree, and, for anotherembodiment, a tree in which edges are labeled with numerals or numericaldata, rather than binary data, may be transformed to a binary edgelabeled tree and/or binary node labeled tree.

Furthermore, a tree in which both the nodes and the edges are labeledmay be referred to in this context as a feature tree and may betransformed to a binary edge labeled tree and/or binary node labeledtree. For example, without intending to limit the scope of claimedsubject matter, in one approach, a feature tree may be transformed byconverting any labeled node with its labeled outgoing edge to an orderedpair of labels for the particular node. Using the embodiment describedabove, this tree may then be transformed to a binary edge labeled tree.

In yet another embodiment, for trees in which data labels do notcomprise simply natural numerals, such as, as one example, trees thatinclude negative numerals, such data labels may be converted to anordered pair of numerals. For example, the first numeral may represent adata type. Examples include a data type such as negative, dollars, etc.As described above, such trees may also be transformed to binary edgelabeled trees, such as by applying the previously described embodiment,for example.

As previously described, trees may be employed to graphically representa hierarchy of data or a hierarchy of a set of data. This has beenillustrated in some detail for binary edge labeled trees, for example.As the previous figures, illustrate, however, such graphicalhierarchical representations typically employ two spatial dimensions todepict the relationship among different pieces of data. This may bedisadvantageous in some situations where a one dimensionalrepresentation or arrangement of symbols, such as is employed withalphabetic letters, for example, that are combined to create a linearcollection of successive symbols or notations, such as words, would bemore convenient.

According to an embodiment, a tree may comprise one or more “subtrees”coupled at the root node of the tree. A subtree of a larger tree maycomprise a “subtree root” node other than the root node of the largerand independently have properties of a tree, except that the subtree ispart of the larger tree. In addition to a subtree root node, a subtreecomprises terminal nodes of a larger tree which descend from the subtreeroot node, and edges and intermediate nodes connecting these terminalnodes with the subtree root node. In one embodiment, the subtree rootnode may be connected to the root node of the larger tree by a singleedge. In another embodiment, the subtree node may be coupled to the rootnode of the larger tree by two or more edges and one or moreintermediate nodes coupled between the root node of the larger tree andthe subtree root node. While a subtree may comprise a portion of alarger tree, the size and shape of the subtree may express informationlike that of a tree having the same size and shape as the subtree.Subtrees coupled together at the root node of a larger tree may bereferred to as “subtree children” of the root node where a subtree maybe referred to as a “subtree child” of the tree in this embodiment.

According to an embodiment, a tree may comprise one or more “partialsubtrees” (PSTs) representing at least a portion of the hierarchicaldata represented by the tree. Here, a PST may a subset of the nodes of alarger tree. This subset of nodes is connected by edges and a singlenode of the subset of nodes provides a “root” node of the PST. In aparticular embodiment, a PST may represent a set of data and/or otherinformation in a hierarchical fashion and may be represented as afinite, rooted, connected, unordered and acyclic graph. Accordingly, aPST has properties of a tree as illustrated above with reference toFIGS. 1-7, except that the PST is a part of a larger tree. As such, in aparticular embodiment, any labels associated with nodes and edges in thefull tree may also be associated with corresponding nodes and edges inany component PST. Also, in a particular embodiment, such a componentPST may be represented by a natural numeral according to an associationof natural numerals and trees such as that illustrated above withreference to FIG. 4, for example. However, this is merely an example ofhow a PST may be associated with a numeral according to a particularassociation embodiment and claimed subject matter may cover associationsof PSTs to numerals according to other association embodiments. Unlike asubtree, a PST need not comprise terminal nodes of the larger tree.According to an embodiment, a PST may comprise a “rooted PST” (RPST)having properties of a PST with the additional feature that the rootnode of the RPST is the same as the root node of the larger tree.

Since a tree is finite, there are a finite number of paths between aroot node of the tree and other nodes in the tree. Similarly, there area finite number of combinations of paths between the root node of a treeand individual ones of the other nodes in the tree. Accordingly, in aparticular embodiment, a finite number of RPSTs may be enumerated from atree having a root node. Numerals may be associated with the enumeratedRPSTs based, at least in part, on an association between trees andnatural numerals such as, for example, illustrated above with referenceto FIG. 4. However, this is merely an example of how RPSTs may beassociated with numerals according to a particular embodiment andclaimed subject matter is not limited in this respect.

According to an embodiment, the enumerated RPSTs of a tree may berepresented as a “set” containing a collection of unordered elements. Ina particular embodiment, the elements of the set of enumerated RPSTs maycontain as elements natural numerals representing individual ones of theenumerated RPSTs according to the aforementioned association betweentrees and numerals. The elements of such a set may be alternativelyexpressed as graphical representations of the individual ones of theenumerated RPSTs. In a particular embodiment, a one-to-one mapping mayrelate elements of the set of RPSTs expressed as natural numerals andelements of the set of RPSTs expressed as graphical representations.Here, such a mapping may enable transforming graphical representationsof RPSTs to corresponding natural numerals and manipulation of suchnatural numerals to provide resulting natural numerals. The resultingnatural numerals may then be transformed back to graphicalrepresentations. However, these are merely examples of how a set ofenumerated RPSTs may be expressed and claimed subject matter is notlimited in these respects.

The term “depth” referred to herein in connection with a tree means thelongest separation between the tree's root node and a terminal node ofthe tree. In a particular embodiment, although claimed subject matter isnot limited in this respect, a depth may be quantified as a number ofsuccessively descending nodes are connected to the root node. In tree100 of FIG. 1, for example, root node 105 is separated from terminalnode 125 by three successively descending nodes, nodes 110, 120 and 125.However, quantifying a depth of a tree by a number of nodes between theroot node and the furthest terminal node of the tree is merely anexample of how depth may be quantified for a particular embodiment, andclaimed subject matter is not limited in these respects.

As illustrated above, a tree is capable of storing, expressing and/orrepresenting hierarchical data. As such, in particular embodiments oftrees, it may be desirable to locate and/or detect particular elementsand/or aspects of hierarchical data which is capable of being stored,expressed and/or represented in a tree. In one particular example, atree may store, express and/or represent hierarchical data that is in adocument that may comprise a piece of information that is of interest.In another embodiment, a tree may store, express and/or representfeatures of a pattern as hierarchical data where one or more features ofthe pattern may be of interest. Techniques for locating and/or detectinghierarchical data in a tree may include, for example, executing aprocess to traverse nodes of a tree that is stored in a memory. Suchtechniques may be computationally intensive.

According to an embodiment, a “target tree” may comprise hierarchicaldata that may or may not include a particular piece of information ofinterest. An embodiment of a process may represent portions of the treeas one or more “target numerals” according to an association of treesand numerals. The particular piece of information of interest may berepresented as a probe numeral that is compared with the one or moretarget numerals to determine a match. In one embodiment, althoughclaimed subject matter is not limited in this respect, such a match mayindicate a presence of a particular piece of information that is ofinterest.

In a particular embodiment, although claimed subject matter is notlimited in this respect, a piece of information of interest may bestored, expressed and/or represented as a “probe tree” having nodesand/or edges and having properties of a tree. As such, a probe tree maybe associated with a “probe numeral” according to an association oftrees and numerals such as that illustrated above with reference to FIG.4, for example. In particular embodiments, the piece of information ofinterest may also be associated with a probe numeral irrespective of anyrepresentation and/or expression of the piece of information of interestas a probe tree. In one particular embodiment, although claimed subjectmatter is not limited in this respect, a probe tree and/or probe numeralmay represent a query for locating and/or detecting a piece ofinformation of interest that is represented in a tree. However, this ismerely an example of how a probe tree and/or probe numeral may beapplied in a particular embodiment and claimed subject matter is notlimited in this respect.

FIG. 8 is a flow diagram illustrating a process embodiment 800 to detectand/or locate information in hierarchical data of a target treeaccording to an embodiment. Block 802 represents a target tree as one ormore numerals according to an association of trees and numerals.According to a particular embodiment of block 802, a computing deviceand/or computing platform may be used to represent the target tree asone or more target numerals according to an association of trees andnumerals. Here, for example, such a computing device and/or computingplatform may comprise one or more processors with a limited native wordsize. In executing processes to determine the target numerals, the oneor more processors may also comprise limited memory. Also, as describedin the aforementioned U.S. patent application Ser. No. 11/005,859,particular embodiments of enumerating numerals associated with trees mayemploy the use of non-composite numerals. Implementations of particularprocesses to enumerate such numerals associated with trees may only becapable of generating non-composite numerals up to a predeterminedmaximum non-composite numeral. Accordingly, a particular implementationof process embodiment 800, although claimed subject matter is notlimited in this respect, may entail partitioning the target tree intosmaller trees comprising clusters and/or subsets of the nodes making upthe larger target tree. Block 802 may then determine the one or moretarget numerals based, at least in part, on numerals associated with thesmaller trees according to an association of trees and numerals.

Block 804 represents information of interest as a probe numeral. Here,the information of interest may be represented and/or expressed as aprobe tree as discussed above. Accordingly, block 804 may represent theinformation of interest as a probe numeral that is associated with theprobe tree according to an association of trees and numerals such asthat illustrated above with reference to FIG. 4, for example. However,this is merely an example of how a probe tree may be associated with anumeral according to a particular association embodiment, and claimedsubject matter is not limited in these respects. Having properties of atree, according to a particular embodiment, a probe tree may comprisenodes which are connected by edges in addition to node and/or edge labelvalues. In particular embodiments, a target tree may comprise largeamounts of data and be much larger than a probe tree. However, this ismerely a comparison of a target tree and a probe tree in a particularembodiment and claimed subject matter is not limited in these respects.Further, a probe tree comprises a root node, and may comprise one ormore terminal nodes. Although claimed subject matter is not limited inthis respect, block 804 may employ techniques illustrated in theaforementioned U.S. patent application Ser. No. 11/005,859 to associatea probe tree with a probe numeral according to an association of treesand numerals.

Block 806 compares the probe numeral determined at block 804 with theone or more numerals representing the target tree to find a match. If amatch is found, process 800 may determine that the target tree comprisesthe information of interest that is represented by the probe numeral.

FIG. 9 is a flow diagram illustrating a process embodiment 850 torepresent a target tree as one or more numerals according to anembodiment of the process embodiment illustrated in FIG. 8. According toan embodiment, although claimed subject matter is not limited in thisrespect, process 850 may generate target numerals for an array. Block806 (FIG. 8) may then compare the probe numeral with target numerals inthe array for determining a match. Block 852 identifies smaller trees inthe target tree. Process embodiment 850 generates the target numeralsfor the array through sequential iteration of blocks 854 through 862.Here, for each smaller tree identified at block 852, block 856 mayenumerate possible RSPTs of the smaller tree as illustrated below withreference to a particular embodiment and in U.S. Provisional PatentApplication No. 60/640,427, titled “Enumeration of Rooted PartialSubtrees” by Karl Schiffmann, J. J. LeTourneau and Mark Andrews, filedon Dec. 30, 2004. Block 858 may then associate the enumerated RPSTs withnumerals according to an association of trees and numerals asillustrated below and in the aforementioned U.S. Provisional ApplicationNo. 60/640,427. Numerals determined at block 858 may then be combinedwith numerals in the array of target numerals at block 860 to provide anupdated array.

Returning to block 852 for identifying smaller trees of a target tree,in a particular example a target tree 900 shown in FIG. 10A comprisessmaller trees A through E as shown in FIG. 10A and in FIG. 10B. In thisparticular embodiment, although claimed subject matter is not limited inthis respect, tree 900 is shown comprising a BELT. However, this ismerely an example of a tree of a particular type and/or configurationfor a target tree, and other target trees may comprising different typesand/or configurations such as, for example, node labeled trees,unlabeled and/or other edge labeled trees. In this particularembodiment, for an individual node in target tree 900 block 852 mayidentify smaller trees as comprising the individual node and nodesdescending from the individual node up to a predetermined depth.

As illustrated below with reference to FIG. 10A, a larger target tree isexpressed as a plurality of smaller trees according to a depthdescending from root nodes from the smaller trees. In this particularembodiment, smaller tree may comprise a root node of the smaller treesand nodes descending from the root node up to a predetermined depth. Inother embodiments, however, a target tree may be expressed as aplurality of smaller trees using other techniques. In a particularembodiment of an ordered target tree, for example, smaller treesexpressing the target tree may be determined based, at least in part, onan order of nodes in the target tree. Again, these are merely examplesof how a target tree may be expressed as a plurality of smaller treesand claimed subject matter is not limited in these respects.

In the particular embodiment illustrated with reference to FIG. 10A,although claimed subject matter is not limited in this respect, asmaller tree identified at block 852 may comprise the individual nodeand nodes descending from the individual node down to a depth of two.For example, smaller tree A comprises root node 902, nodes descendingfrom root node 902 at a depth of one (nodes 904, 906 and 908) and nodesdescending from root node 902 descending at a depth two (nodes 910, 912,914 and 916), but does not comprising nodes descending at a depth ofthree (nodes 920 through 928). Block 852 may similarly identify smallertrees corresponding with other nodes in target tree 900 comprising atleast one child node. In this particular example, block 852 may identifysmaller tree B comprising node 906 as its root node, smaller tree Ccomprising node 910 as its root node, smaller tree D comprising node 908as its root node and smaller tree E comprising node 916 as its rootnode.

It should be observed that smaller trees C and E, having nodes 910 and916 as respective root nodes, comprise nodes descending from a root nodeat a depth of one but no nodes descending at a depth of two.Accordingly, for an individual node having at least one descendant childnode, block 852 may identify a smaller tree comprising nodes descendingdown to the predetermined depth, to the extent that there are nodesdescending from the individual node at the predetermined depth.

As shown in FIG. 10A, according to a particular embodiment, smallertrees A through E are shown as overlapping in that some of the nodes intarget tree 900 are a part of more than one of the smaller trees Athrough E. In this particular embodiment, smaller trees A through Ecomprise PSTs of target tree 900. Accordingly, possible PSTs of targettree 900 up to a particular depth (a depth of two in this particularexample) may be identified as smaller trees in block 852. In aparticular embodiment, although claimed subject matter is not limited inthis respect, a depth of smaller trees may be based, at least in part,on an expected and/or maximum depth of a probe tree. Here, for example,a depth of smaller trees may be selected as a maximum depth of a probetree to enable matching the probe tree with a smaller tree. Accordingly,in this particular embodiment, a probe numeral associated with such amaximum depth probe tree may be capable of matching one or more targetnumerals representing the target tree at block 806. FIG. 11 shows probetrees 950 and 952 comprising a depth of two. By identifying smallertrees of target tree 900 having up to a depth of two thus enablesfinding a match of probe numerals (associated with respective probetrees 950 and/or 952) with one or more numerals associated with RPSTs ofsmaller trees A through E.

It should be understood that the particular predetermined depth used foridentifying smaller trees in this particular example is arbitrary andthat other depths of one or greater may be chosen. In one particularembodiment, although claimed subject matter is not limited in thisrespect, a particular depth may be chosen based, at least in part, onresources available at a computing device and/or computing platform forenumerating RPSTs at block 856 and associating numerals with enumeratedRPSTs at block 858. Here, for example, blocks 856 and/or 858 may becapable of performing these operations for a smaller tree having ashorter depth with fewer computing resources than for a smaller treehaving a longer depth. On the other hand, as discussed above, theparticular predetermined depth used for identifying smaller trees mayalso be based, at least in part, on an expected size of a probe tree.

While nodes of a smaller tree may be determined based, at least in part,on a depth descending from a root node of the smaller tree in particularembodiments, such nodes of a smaller tree may be determined using othertechniques and claimed subject matter is not limited in this respect. Inone alternative embodiment, for example, one or more nodes of a smallertree may be determined, at least in part, on node and/or edge labelvalues associated with the one or more nodes. Here, for example, a nodedescending from a root node of a smaller tree may determined to beincluded in the smaller tree based, at least in part, on a node and/oredge label value associated with the descending. Again, this is merelyanother example of how descending nodes may be determined to be includedas part of the smaller tree and claimed subject matter is not limited inthis respect.

Process embodiments 800 and 850 will be illustrated by way of examplewith reference to FIGS. 10A, 10B and 11. Here, tree 900 comprises atarget tree which expresses and/or represents hierarchical data. Trees950 and 952, in this particular example, are probe trees representingpieces of information that are of interest. Block 802 represents thetarget tree 900 as one or more target numerals. Here, according to aparticular embodiment, process embodiment 850 may identify smaller treeshaving a depth of two as A through E of the target tree 900 asillustrated above with reference to FIG. 10. Here, as illustrated inFIG. 10B, these smaller trees A through E may be associated withnumerals according to the association of BELTs and numerals illustratedabove with reference to FIG. 4 as follows:

A 6586 B 249 C 12 D 186 E 6However, it should be understood that these are numerals associated withsmaller trees A through E according to a particular association of treeswith numerals and that claimed subject matter is not limited in thisrespect.

For a particular embodiment, a “full tree” is defined as an integraltree comprising all of its nodes, edges coupling the nodes to oneanother and any labels associated with the nodes or edges. Therefore, afull tree includes all of its nodes and elements completely connected.Also, such a full tree may be represented by a numeral denoted here as“FT” according to an association of trees and numerals. The notation“{RPSTs:: FT}” provides a shorthand notation for this particularembodiment to indicate the set of unique, unordered RPSTs that may beformed from a full tree “FT.” In one embodiment, the elements of{RPSTs:: FT} may comprise natural numerals representing correspondingmember RPSTs.

Returning to process embodiment 850, in this particular example, blocks854 through 862 may generate an array of numerals comprising numeralsassociated with RPSTs of smaller trees A through E. In particular, basedat least in part on numerals associated with respective smaller trees Athrough E according to a particular association of trees and numerals,blocks 854 through 862 may form an array of target numerals to comprise{RPSTs:: 6586} U {RPSTs:: 249} U {RPSTs:: 12} U {RPSTs:: 186} U {RPSTs::6}. The discussion below with reference to FIGS. 8 through 18 describesa particular embodiment of a process for enumerating RPSTs to determinemember numerals for the individual expressions {RPSTs:: 6586}, {RPSTs::249}, {RPSTs:: 12}, {RPSTs:: 186} and {RPSTs:: 6} corresponding withsmaller trees A, B, C, D and E, respectively. The discussion below withreference to FIGS. 19 through 28 illustrates a particular numericalexample for determining member numerals of {RPSTs:: 249} correspondingwith smaller tree B in this particular embodiment. It should beunderstood that the process embodiment described with reference to FIGS.8 through 18 may be similarly applied to determine member numerals of{RPSTs:: 249} and the remaining expressions corresponding with smallertrees A, C, D and E as follows:

A: {RPSTs:: 6586}={2, 3, 6, 7, 9, 13, 14, 18, 19, 21, 26, 37, 38, 39,42, 49, 57, 74, 78, 89, 91, 98, 111, 114, 133, 169, 178, 182, 222, 247,259, 266, 267, 338, 481, 494, 518, 534, 623, 703, 962, 1157, 1246, 1369,1406, 2314, 2738, 3293, 6586};

B: {RPSTs:: 249}={2, 3, 5, 6, 11, 15, 17, 31, 33, 51, 83, 93, 249};

C: {RPSTs:: 12}={2, 3, 4, 6, 12};

D: {RPSTs:: 186}={2, 3, 4, 5, 6, 10, 11, 12, 15, 22, 30, 31, 33, 62, 66,93, 166}; and

E: {RPSTs:: 6}={2, 3, 6}.

Probe trees 950 and 952, shown in FIG. 11, have a depth of twocorresponding with the maximum depth of the smaller trees A through E.Returning to process embodiment 800 in this particular example, block804 represents probe trees 950 and 952 (FIG. 11) as probe numerals.Here, probe trees 950 and 952 may be associated with numerals “31” and“26”, respectively, according to an association of trees and numerals asillustrated with reference to FIG. 4 in this particular example(although probe trees 950 and 952 may be associated with differentnumerals according to a different association between trees andnumerals). Accordingly, probe numerals determined at block 804 maycomprise “31” and “26” in this particular embodiment. Here, block 806may determine a match of probe numeral “26” with a target numeralassociated with an RSPT of smaller tree A, and determine a match ofprobe numeral “31” with a target numeral associated with an RSPT ofsmaller trees B and D. A match of probe numeral “26” may be graphicallyobserved as a match of probe tree 952 with the PST of target tree 900(in smaller tree A) formed by nodes 902, 904, 908 and 914, and binarylabeled edges connecting these nodes. Similarly, a first match of probenumeral “31” may be graphically observed as a match of probe tree 950with a first PST of tree 900 (in smaller tree B) formed by nodes 906,910, 920 (and/or 922) and 924, and binary labeled edges connecting thesenodes. A second match of probe numeral “31” may be graphically observedas a match of probe tree 950 with a second PST of tree 900 (in smallertree D) formed by nodes 908, 916, 926 and 928, and binary labeled edgesconnecting these nodes.

The embodiments described above relate to, among other things, detectinga presence of a piece of information of interest in hierarchical datarepresented and/or expressed as a target tree. In other embodiments,although claimed subject matter is not limited in this respect, alocation of the detected piece of information of interest may bedetermined. Here, particular elements of a target tree (e.g., nodes,edges and/or smaller trees) may themselves be associated with particulardata locations in a hierarchical database, for example. As illustratedabove, target numerals may be associated with particular smaller treesformed from a larger target tree. In addition to finding a match betweena probe numeral and a target numeral generated from any of the smallertrees, process 800 may also provide an indication of which smaller treeproduced matching target numeral(s). In the example above, for thepurposes of illustration, probe numeral “26” matches a target numeralfrom smaller tree A and probe numeral “31” matches target numerals fromsmaller trees B and D. With smaller tree A being associated with aparticular location in a database, in a particular example, matchingprobe numeral “26” with a target numeral from smaller tree A mayindicate a location of information of interest in a location of thedatabase associated smaller tree A. Similarly, with smaller trees B andD being associated with particular locations in a database, in aparticular example, matching probe numeral “31” may indicate a locationof information of interest in locations of the database associated withsmaller trees B and D.

In one particular example of the above described process applied tobiometric pattern recognition, although claimed subject matter is notlimited in this respect, a known biometric pattern (e.g., facialfeatures) may be modeled as hierarchical data in a target tree. One ormore detected features of a subject or specimen may be modeled ashierarchical data in a probe tree. One or more target numerals mayrepresent the known biometric pattern and a probe numeral may representthe detected features. A comparison of the target numerals with theprobe numerals may then indicate whether the one or more detectedfeatures are present in the known biometric pattern upon detection ofone or more matches.

In another particular example, the technique described above may be usedfor processing queries to an extensible markup language (XML) documentor set of documents. Here, hierarchical data in an XML document or setof documents may be expressed and/or represented as one or more targettrees. A query to the one or more XML documents may be represented as aprobe tree. Applying the techniques above, according to a particularembodiment, the target tree representing the one or more XML documentsmay be represented as one or more target numerals and the probe tree maybe represented as a probe numeral. The probe numerals, representing theprobe tree and the query, may then be compared with the one or moretarget numerals, representing the target tree and hierarchical data inthe one or more XML documents. Detecting a match with a probe numeralwith one or more of the target numeral may indicate a presence ofinformation in the one or more XML documents that is the subject of thequery. However, this is merely a particular example of an application ofthe above described process and claimed subject matter is not limited inthis respect.

Embodiments described above with reference to FIGS. 8 through 11 employa particular example which represents and/or expresses hierarchical datain the form of a BELT. However, process embodiments 800 and 850 mayemploy representations of hierarchical data in other types of treeswhich are associated with numerals according to an association of treesand numerals. In other embodiments, for example, a target tree and/orprobe tree may comprise node labeled trees, trees with neither edge nornode labels, or m-ary edge labeled trees having edges capable ofstoring, holding and/or representing three or more possible values.Regardless of a particular form of a target tree and/or probe tree inany particular embodiment, a target tree may be represented as one ormore target numerals according to an association of trees and numerals.Information of interest, represented by a probe tree in the particularform for example, may similarly be represented as a probe numeral forcomparison with the one or more target numerals.

FIGS. 12 through 31 illustrate an embodiment of a process to enumerateRPSTs of a tree (e.g., a smaller PST of a target tree as illustratedabove). As shown in FIGS. 12 through 15, where FT represents tree 1200,elements of {RPSTs:: FT} comprise the component RPSTs shown in FIGS. 13,14 and 15 (among other component RPSTs as illustrated below). FIG. 12 isa schematic diagram of a tree 1200 illustrating, by way of example,RPSTs of tree 1200. Tree 1200 comprises a root node 1202 and nodes 1204,1206, 1208, 1210 and 1212 coupled to the root node 1202 by edges andintermediate nodes. In the presently illustrated embodiment, tree 1200comprises a BELT. It should be understood, however, that other types oftrees (either labeled trees or unlabeled structure trees) may similarlycomprise RPSTs and that claimed subject matter is not limited in thisrespect.

FIGS. 13, 14 and 15 are schematic diagrams of some RPSTs of tree 1200according to an embodiment. These RPSTs include root node 1202 of tree1200, at least one other node of tree 1200, and any edges orintermediate nodes in tree 1200 coupling the at least one other node tothe root node 1202. However, it should also be understood that the setof RPSTs of tree 1200 may also include, for example, a single node 1202or the full tree 1200. FIGS. 13 and 14 show RPSTs including root node1202 and one other node in tree 1200 which connected via an edge in tree1200 between root node 1202 and the one other node. FIG. 15 shows anRPST including root node 1202, nodes 1208, 1210 and 1212, andintermediate node 1206 coupling the root node 1202 to the nodes 1208,1210 and 1212. It should be understood that FIGS. 13, 14 and 15 aremerely examples of RPSTs that may be formed from tree 1200, and thatthese examples are not intended to provide an exhaustive enumeration ofRPSTs that may be formed from tree 1200.

FIG. 16 is a flow diagram illustrating a process 1250 to enumerate RPSTsfrom a tree according to an embodiment. In this particular embodiment,the process 1250 may enumerate the RPSTs of full tree FT to defineelements of {RPSTs:: FT} as a result. FT may be represented as a naturalnumeral received at block 1252. {RPSTs:: FT} may be initialized as theempty set at block 1254 to be subsequently filled with elementsrepresenting RPSTs enumerated in subsequent portions or process 1250.

According to an embodiment, the process 1250 recognizes that the fulltree may represent any one of four different configurations: an emptytree; a single node tree; a tree comprising a single subtree connectedto a root node of the full tree by an edge; and two or more subtreesconnected to the root node of the full tree by respective edges.Accordingly, the process 1250 enumerates the RPSTs of the full treebased, at least in part, on the particular configuration of the fulltree. Diamond 1256 determines whether FT represents an empty treecontaining no nodes. If so, {RPSTs:: FT} remains defined as the emptyset and process 1250 terminates at block 1268. If diamond 1258determines that FT contains a single node tree, block 1260 updates{RPSTs:: FT} to include a natural numeral expressing a single node tree(here, {r}).

At diamond 1262 through block 1268, process 1250 enumerates RPSTs based,at least in part, on the configuration of the full tree as having eithera single subtree connected to the root node of the full tree by an edge,or two or more subtrees connected to the root node by respective edges.If FT represents a single subtree connected to the root node of the fulltree by an edge, block 1264 enumerates the RPSTs of the single subtree.Here, the RPSTs of the full tree may be determined, at least in part,from the RPSTs of the single subtree.

If FT represents a full tree having two or more subtrees connected tothe root node of the tree by respective edges, block 1266 may enumeratethe RPSTs of the individual ones of the two or more subtrees. At leastsome of the RPSTs of the full tree may be determined, at least in part,from RPSTs of the individual subtrees. Block 1266 may then enumerateadditional RPSTs of the full tree based, at least in part, combinationsof the enumerated RPSTs merged at the root node of the full tree.

According to an embodiment, blocks 1264 and 1266 may be carried out byrecursive execution of at least a portion of the process 1250. At block1264, for example, the single subtree of the full tree may itselfcomprise two or more subtree children connected by respective edges to anode. Block 1264 may execute portions of block 1266 to enumerate theRPSTs of the subtree based, at least in part, on RPSTs enumerated fromindividual ones of the subtree children of the single subtree.Similarly, block 1266 may enumerate RPSTs of individual ones of thesubtrees connected to the root node of the full tree by executingportions of block 1264.

FIG. 17 is a schematic diagram illustrating a representation anembodiment of a subtree comprising an edge having a label “e” with asubtree root node having a label “n” where “e” and “n” may berepresented by discrete values (e.g., Boolean, binary, integer, naturalnumeral and/or whole numeral values). A value associated with a “childtree” of the subtree may be represented by “x” which may representinformation expressed as a natural numeral according to an associationof natural numerals with trees as illustrated above with reference toFIG. 4. Like a tree, subtree and RPST, such a child tree may haveproperties of a tree and be associated with a natural numeral accordingto an association between trees and natural numerals. However, this ismerely an example of a representation of a subtree and its child tree,and claimed subject matter is not limited in this respect.

As described below in connection with relation (1), a push operation maydefine a relationship between a subtree and a child tree of the subtree.As an association between trees and natural numerals may associateparticular trees with natural numerals (e.g., as illustrated in FIG. 4),a push operation may define a relationship between a natural numeralassociated with a subtree and a natural numeral associated with a childtree of the subtree. Similarly, a push operation may also define arelationship between natural numerals representing RPSTs of the childtree and natural numerals associated with at least some of the RPSTs ofthe subtree. A value of, or natural numeral associated with, the subtreeshown in FIG. 17 may be expressed as the result of a push operation onthe child tree having the value x. Such a push operation on the childtree may be represented in relation (1) as follows:push(j,k,x)=P[kx+j−k+(2−r)], if j<k and k>0  (1)where:

-   -   P(m)=Kleene enumeration function for generating a sequence of        non-composite numerals illustrated with reference to FIG. 5;    -   k=total number of values possible for a label;    -   j=actual computed label index value;    -   x=value of, or natural numeral associated with, “pushed” child        tree; and    -   r=defined value of tree system root/single node tree (e.g.,        either 0 or 1).

It should be understood that while the push operation of relation (1) issuitable for performing specific embodiments described herein, this pushoperation is merely an example of how a push operation may be performedand claimed subject matter is not limited in this respect. Additionally,it should be noted that the value of “r” is selected based upon aparticular association of natural numerals and trees according to anassociation embodiment. Here, such an association of natural numeralsmay define a particular natural numeral to represent a tree comprising asingle node. In the association of natural numeral with trees of FIG. 4,for example, the single node is associated with “1” defining r=1.However, this is merely an example of how a natural numeral mayrepresent a single node for a particular association embodiment andclaimed subject matter is not limited in this respect.

It should also be noted that “j” (the actual computed label index valueassociating the root node with the pushed subtree) is a function of thespecific values of “e” (the specific edge label) and “n” (the specificnode label). In the particular case of a BELT, for example, there may beno node values such that “j”=“e”. The value of “k” (total number ofpossible index values) may be determined as function of thepossibilities of values of “e” (edge label value) and “n” (node labelvalue) and, in a particular embodiment, “k” may be determined as thenumber of possibilities for “e” multiplied by the number ofpossibilities for “n.” Again, in the particular case of a BELT, “k”equals the number of possibilities for the value “e” since there are nonode labels.

The techniques described herein for enumerating RPSTs of a full tree maybe applied to any particular type of tree. For illustration purposes,particular examples described herein are directed to enumerating RPSTsof a BELT. Accordingly, while it is understood that an actual computedindex value associating the root node with the pushed subtree may bedetermined from node labels (having a value “n”) and/or edge labels(having a value “e”), for simplicity the remaining discussion willdenote the actual computed label index value “j” as an edge label valueof an edge connecting a root node of a tree to a pushed child tree.

In enumerating at least some RPSTs of a tree based, at least in part, onenumerated RPSTs of a subtree of the RPST, it may be useful to express apush operation on multiple RPSTs in a single push operation. In additionto applying a push operation to a tree having a value x, the pushoperation may be applied to multiple trees or tree elements of a set(here, an unordered collection of elements representing trees, RPSTs,subtrees and/or child trees of a subtree) in relation (2) as follows:push[j,k,{a,b,c}]={push(j,k,a)}U{push(j,k,b)}U{push(j,k,c)}  (2)where a, b and c are numerical representations of tree elements in thepushed set. The result of the operation of relation (2) may be referredto as a “pushed set” of tree elements.

FIGS. 18 and 19 illustrate applications of the push operation ofrelation (1) to specific subtrees. FIG. 18 shows a BELT having a valueof “2” according to the association of trees and natural numerals shownin FIG. 4. As such, a push operation on this tree would define x=2, k=2and r=1. FIG. 19 illustrates the result of a push of the tree in FIG. 18by an edge having a label zero (i.e., a zero-push). The value of j forthis push operation is zero. Accordingly, the push operation provides anumeral associated with the pushed BELT as follows:push(j,k,x)=P[2*2+0−2+2−1]=P[3]=5.It should be understood, however, the application of the push operationof relation (1) to a BELT as illustrated in FIGS. 18 and 19 are merelyparticular examples of the push operation and the push operation may besimilarly applied to non-BELT trees.

To enumerate RPSTs of a subtree of a full tree, it may be useful todetermine a numeral associated with a child tree of the subtree based,at least in part, on a numeral associated with the subtree (the naturalnumerals being based, at least in part, on an association between treesand numerals). Like the push operation of relation (1), according to anembodiment, an “inverse push” operation may define a relationshipbetween a subtree (e.g., a subtree of a parent full tree) and the childtree of the subtree (as illustrated in FIG. 13). Here, such an inversepush operation may define a relationship between numerals associatedwith such subtrees and numerals associated with respective child treesof such subtrees. Operating on such a natural numeral associated with asubtree (according to an association of trees with natural numerals asillustrated with reference to FIG. 4, for example), an inverse pushoperation may provide a natural numeral associated with a child treeportion. In a particular embodiment, an inverse push operation mayprovide as a result as (1) a natural numeral representing a child tree(“Child”) and an edge label value linking the child tree with the rootnode of the parent full tree (“j”). For example, an inverse pushoperation on a tree may be represented in relation (3) as follows:push⁻¹(r,k,ST)=<Child,j>Child=Int[(P ⁻¹(ST)+k−(2−r))/k]; andj=[P ⁻¹(ST)+k−(2−r)]modulo[k]  (3)where:

-   -   P⁻¹(h)=an inverse of the Kleene enumeration function for        generating a sequence of non-composite numbers illustrated with        reference to FIG. 5;    -   ST=value of, or natural numeral associated with, subtree with        edge label value “j”;    -   Child=natural numeral representing child tree of subtree        represented by ST;    -   j=edge label value linking the child tree with the root node of        parent full tree;    -   k=total number of possible edge label index values linking the        child tree with the root node of parent full tree; and    -   r=defined value of tree system root/singleton node (either 0 or        1).

It should also be understood that the inverse push operation of relation(3) is merely an example of an inverse push operation used to determinea natural numeral associated with a child tree based, at least in parton a natural numeral associated with a parent subtree, and that claimedsubject matter is not limited in this respect. For example, forsimplicity relation (3) assumes that information of a computed indexvalue “j” associating the root node of the parent full tree and thechild tree may be derived from edge label values in the absence of nodelabel values (e.g., as in the case of a BELT). However, relation (3) maybe expanded to apply to other non-BELT trees. Applied to the tree ofFIG. 17, for example, the inverse push operation of relation (3) may beexpanded to yield a natural numeral representing the child tree as thevalue “x” and the label index value j (being a function of edge and nodelabel values “e” and “n”).

In the particular embodiment of an inverse push operation illustrated inrelation (3), the inverse Kleene enumeration function, P⁻¹(h), providesa result based upon ST (value of, or natural numeral associated with thesubtree). Since the Kleene enumeration function generates non-compositenatural numerals, the domain of P⁻¹(h) may be limited to non-compositenatural numerals. In connection with the association of natural numeralsand trees illustrated with reference to FIG. 4, accordingly, the inversepush operation of relation (3) may be applied to natural numeralsrepresenting trees having a root node coupled to a single node (orhaving a single subtree connected to the root node). In one particularimplementation of the inverse of the Kleene enumeration function,P⁻¹(h), a look-up table may associate values of h and P⁻¹(h) as shown inTable 1 as follows:

TABLE 1 h P⁻¹(h) 2 1 3 2 5 3 7 4 11 5 13 6 17 7 19 8 23 9 29 10 31 11 3712 41 13 43 14 47 15 53 16 59 17 61 18 67 19 71 20 73 21 79 22 83 23

To enumerate at least some of the RPSTs of a full tree having two ormore subtrees connected to the root node of FT by different edges, itmay be useful to determine combinations of RPSTs enumerated from thedifferent subtrees. In the case of a full tree comprising two subtreesconnected to the root node by two edges, in a particular example,individual elements of a first set of RPSTs of the full tree derivedfrom a first subtree (denoted as “X” for the purposes of illustration)may be combined or merged with individual elements of a second set ofRSPTs of the tree derived from a second subtree (denoted as “Y” for thepurposes of illustration). Here, the elements of X and Y may representindividually enumerated RPSTs of the tree derived from the first andsecond subtrees, respectively. In a particular embodiment, the elementsof X and Y may be represented as natural numerals associated withenumerated RPSTs derived from the respective first and second subtrees(according to an association of trees and natural numerals asillustrated in FIG. 4, for example). Accordingly, a merger of an RPSTrepresented by an element in X with an RPST represented by an element inY at the root node of the tree may be represented by a multiplication ofthese natural numerals resulting in a natural numeral representing theRPST resulting from the merger.

According to one embodiment, a merger operation discussed above (e.g.,for combining trees at their root nodes to provide a graphical andnumerical expression of the resulting merged trees) may be expanded tocreate a set merger operation to include a merger among RPSTs (e.g.,derived from different subtrees as illustrated above). Here, a memberRPST of a first set merges with a member RPST of a second set to providea member of a third, merged set containing the merged RPSTs as elements,for all members of both first and second sets. Regarding theaforementioned representation of the RPSTs as natural numerals, the setmerger operation to merge sets X and Y may be expressed as follows:

$\begin{matrix}\begin{matrix}{{XxY} = {{\left\{ {x_{1},x_{2},x_{3},\ldots\mspace{14mu},x_{n}} \right\} \times \left\{ {y_{1},y_{2},y_{3},\ldots\mspace{14mu},y_{m}} \right\}} =}} \\{= {\left\{ {{x_{1}*y_{1}},{x_{1}*y_{2}},{x_{1}*y_{3}},\ldots\mspace{14mu},{x_{1}*y_{m}}} \right\} U\left\{ {{x_{2}*y_{1}},{x_{2}*y_{2}},} \right.}} \\{\left. {{x_{2}*y_{3}},\ldots\mspace{14mu},{x_{1}*y_{m}}} \right\} U\left\{ {{x_{3}*y_{2}},{x_{3}*y_{2}},{x_{3}*y_{3}},\ldots\mspace{14mu},{x_{3}*y_{m}}} \right\}} \\{U\mspace{14mu}\ldots\mspace{14mu} U\left\{ {{x_{n}*y_{1}},{x_{n}*y_{2}},{x_{n}*y_{3}},\ldots\mspace{14mu},{x_{n}*y_{m}}} \right\}}\end{matrix} & (4)\end{matrix}$where:

x₁, x₂, X₃, . . . X_(n) are the natural numerals representing RPSTs inset X; and

y₁, y₂, y₃, . . . y_(m) are the natural numerals representing RPSTs inset Y;

FIGS. 20, 21 and 22 are flow diagrams illustrating a process toenumerate RPSTs from a full tree which applies the aforementioned pushoperations, inverse push operation and set merger operation of relations(1) through (4) according to a particular embodiment. FIG. 20illustrates a process 1600 to receive a natural numeral representing afull tree, FT, at block 1602. Diamond 1604 may determine whether thetree represented by FT comprises an empty tree. In this particularembodiment, the value “r” is zero or one depending on whether a singlenode tree is associated with a value of zero or one (e.g., depending ona particular association embodiment defined above in connection with thepush operation of relation (1)). Here, diamond 1604 determines whetherthe tree represented by FT is an empty tree based upon whether FT isless than r. However, this is merely an example of a technique toidentify an empty tree and claimed subject matter is not limited in thisrespect. Process 1600 terminates at block 1612 if diamond 1604determines that FT represents an empty tree. Diamond 1606 determineswhether FT represents a single node tree based upon whether FT equals r.However, this is merely an example of a technique to identify a singlenode tree and claimed subject matter is not limited in this respect. IfFT comprises a single node tree, block 1608 assigns r as the RPSTelement of {RPSTs:: FT} and the process 1600 terminates at block 1612.

Diamond 1606 determines whether FT comprises two or more nodes bydetermining whether FT is greater than r. If so, block 1610 may initiateexecution of process 1700 shown in FIG. 21. Block 1704 initializes theset {RPSTs:: FT} to contain no elements while execution of the remainingportions of process 1700 may subsequently add elements to this set. Asdiscussed below, process 1700 may be recursively executed to determine,for example, RPSTs of subtrees of the full tree represented by FT.Accordingly, diamond 1706 determines, much like diamond 1606 of process1600, whether FT (e.g., in a recursive execution) represents a singlenode tree. If so, then process 1700 (and process 1600) terminate atblock 1720.

If FT is greater than r, diamond 1708 determines whether FT represents atree comprising a single subtree (e.g., comprising a child tree pushedfrom the full tree node by an edge as shown in FIG. 17) or a treecomprising a root node that merges two or more subtrees (e.g., as shownin FIG. 12). Here, diamond 1708 determines that FT represents a treecomprising a single subtree if FT comprises a non-composite numeral, anddetermines that FT represents a tree having a root node merging two ormore subtrees if FT comprises a composite numeral. For this embodiment,an association between trees and natural numerals may associatecomposite natural numerals with trees merging two or more subtrees at aroot node, and may associate non-composite numerals with treescomprising a single subtree (here, a pushed child tree coupled to theroot node of the tree by an edge). However, this is merely an example ofan association embodiment, and claimed subject matter is not limited inthis respect.

Similar to the process 1250 illustrated above with reference to FIG. 16,process 1700 employs a process to enumerate the RPSTs of a full treebased, at least in part, on the configuration of the full tree. Here, itshould be observed that block 1710 through 1716 collectively represent aprocess of enumerating RPSTs of a full tree having a single subtreeconnected to the root node of the full tree by a single edge. Process1700 may employ a different process at block 1718 if the full treecomprises two or more subtrees merged at a root node. As illustratedbelow, the RPSTs of a full tree may be determined, at least in part, byan enumeration of RPSTs of child trees of subtrees. Accordingly, theprocess 1700 of enumerating RPSTs of a full tree may be recursivelyexecuted for determining such RPSTs of the child trees.

If diamond 1708 determines that FT represents a tree comprising a singlesubtree connected to a root node by a single edge, block 1710 executesan inverse push operation on FT as illustrated above in relation (3) todetermine a natural numeral “child” representing the child tree coupledto the root node of the tree represented by FT (and edge label value “j”linking the root node with the child tree). At least some of the RPSTsof the tree represented by FT may be derived from RPSTs of the childtree determined at block 1710. Accordingly, block 1712 may recursivelyexecute process 1700 to enumerate the RPSTs of the child tree ({RPSTs::child}). Here, the recursively executed process may apply the naturalnumeral “child” representing the child tree (e.g., as determined atblock 1710) as the FT input value. Block 1714 then combines the singlenode tree represented by “r” with the set of enumerated RPSTs determinedat block 1712. Block 1716 then performs a push operation according torelation (2) on the elements of this combined set {RPSTs:: child} tocomplete the enumeration of the elements of {RPSTs:: FT} in a pushed setwith the edge label value “j” determined from the inverse push operationat block 1710.

If diamond 1708 determines that FT represents a tree comprising a rootnode that merges two or more subtrees, block 1718 may enumerate theelements of {RPSTs:: FT} by executing a process 1800 shown in FIG. 22.As such, block 1718 may provide the composite numeral FT as an inputvalue to process 1800 at block 1802. Subsequent blocks 1804 through 1818may then enumerate RPSTs for individual subtrees merged at the root nodeof the tree represented by FT, and determine {RPSTs:: FT} fromcombinations of the RPSTs enumerated from particular ones of the mergedsubtrees.

A processing loop of blocks 1806 through 1816 incrementally factors thecomposite numeral FT into non-composite numerals “ST” representingindividual subtrees merged at the root node of the tree represented byFT. Again, this particular embodiment includes an association betweentrees and natural numerals that associates composite natural numeralswith trees merging two or more subtrees at a root node and associatesnon-composite numerals with trees having a root node coupled to a singlepushed subtree; however, claimed subject matter is not limited in scopeto this particular embodiment. Here, block 1804 initializes a“remainder” as FT and block 1808 determines the non-composite numeral STas the smallest non-composite factor of the remainder. If the remainderis decreased to below r, representing a single node tree in thisparticular embodiment), sequential execution returns to process 1700 atblock 1818.

Through successive executions of the processing loop of blocks 1806through 1816, block 1808 may sequentially factor the numeral FT intonon-composite numerals representing subtrees of the tree represented byFT and connected to a root node of FT. According to a particularassociation embodiment, these non-composite numerals may representindividual ones of subtrees merged at a root node of the treerepresented by FT. As at least a portion of the RPSTs of the treerepresented by FT may be determined from the RSPTs of these subtrees,block 1810 may recursively execute the process 1700 to enumerate theRPSTs of the subtrees represented by the non-composite values STdetermined at block 1808.

It should be observed that the elements of {RPSTs:: FT} are derived fromthe RPSTs enumerated from individual subtrees (determined through loopiterations of block 1810). In addition to these elements, {RPSTs:: FT}also includes merged combinations of RPSTs derived from RPSTs enumeratedfrom different subtrees at block 1810 in different loop iterations.Through executions of the loop of block 1806 through 1816, block 1812updates {RPSTs:: FT}. By way of example, for the purpose ofillustration, in an initial iteration of the loop, block 1812 may merelyassign elements to {RPSTs:: FT} (which is initialized as the empty set)to include the RPSTs enumerated at block 1810 from a first subtree ofthe tree represented by FT. In a second iteration of the loop, block1810 enumerates RPSTs of a second subtree of the tree represented by FT.In addition to adding the enumerated RPSTs of the second subtree to{RPSTs:: FT} (updated in the initial loop iteration to include RPSTsenumerated from the first subtree), block 1812 in the second iterationalso updates {RPSTs:: FT} to include RPSTs formed from the merger of thecurrent individual elements of {RPSTs:: FT} (again, updated from theinitial iteration) with individual enumerated RPSTs of the secondsubtree. Here, block 1812 employs a set merger operation according torelation (4) to determine a merger of the current individual elements of{RPSTs:: FT} (e.g., assigning the elements of {RPSTs:: FT} to “X”) withthe individual elements of the enumerated RPSTs of the second subtree(e.g., assigning the elements of RPSTs of the second subtree to “Y”).Subsequent iterations of the processing loop of blocks 1806 through 1816may then enumerate the RPSTs of additional subtrees, and update {RPSTs::FT} based upon the elements of {RPSTs:: FT} updated in the previousiteration and the enumerated RPSTs of the subsequent subtree children inlike fashion.

FIGS. 23 through 31 are schematic diagrams of trees illustrating aspecific example of enumerating RPSTs of a tree according to the processembodiments of FIGS. 20, 21 and 22. FIG. 23 shows a tree 1900 whichcomprises a BELT for this particular illustration of an embodiment,however, it should be understood that the processes described forenumerating RPSTs are applicable to non-BELT trees as well, and thatclaimed subject matter is not limited in this respect.

Tree 1900 may be represented as a numeral “249” according to anassociation of trees and natural numerals as described above withreference to FIG. 4, for example. As tree 1900 comprises two or moresubtrees merged at a root node 1902, the natural numeral 249 comprises anon-composite natural numeral. Commencing execution of process 1600 fordetermining {RPSTs:: 249}, block 1602 defines FT=249. Since tree 1900comprises a BELT in this particular embodiment, the value of “r” may be“1” consistent with the application of the push operation of relation(1). Accordingly, diamonds 1604 and 1606 direct initiating execution ofprocess 1700 through block 1610.

Block 1704 initializes {RPSTs:: 249} as an empty set to be subsequentlyfilled with natural numerals representing RPSTs of tree 1900. Since 249(here, FT) comprises a composite natural numeral, block 1718 mayinitiate an instance of process 1800. Block 1804 initializes“remainder”=249 and block 1808 determines ST to be the natural numeral 3(since 249 may be factored into two non-composite numerals 3 and 83).

Block 1810 may initiate a first recursive instance of process 1700 whileproviding FT=ST=3 as an input value, diamond 1708 determines that 3 is anon-composite numeral. Block 1710 performs an inverse push operationaccording to relation (3) to determine a natural numeral representativeof the child tree of the subtree corresponding to the natural numeral 3and an edge label value of an edge linking the child tree with the rootnode as follows:

$\begin{matrix}\begin{matrix}{{{{push}^{- 1}\left( {{r = 1},{k = 2},{{ST} = 3}} \right)} = {< {child}}},{j >}} \\{{child} = {{Int}\left\lbrack {\left( {{P^{- 1}(3)} + 2 - \left( {2 - 1} \right)} \right)/2} \right\rbrack}} \\{= {{Int}\left\lbrack {\left( {2 + 2 - \left( {2 - 1} \right)} \right)/2} \right\rbrack}} \\{= 1} \\{j = \left\lbrack {\left( {{P^{- 1}(3)} + 2 - \left( {2 - 1} \right)} \right\rbrack{{modulo}\lbrack 2\rbrack}} \right.} \\{= \left\lbrack {\left( {2 + 2 - \left( {2 - 1} \right)} \right\rbrack{{modulo}\lbrack 2\rbrack}} \right.} \\{= 1}\end{matrix} & (5)\end{matrix}$

Block 1712 initiates execution of a second recursive instance of process1700, initializing {RPSTs:: child}=Ø and terminating at block 1720through diamond 1706 (since child=1≦r). Returning to block 1714 of thefirst recursive instance of process 1700, {RPSTs:: child} is updated tobe {r}={1} for this particular case of a BELT. Block 1716 then performsa push operation on the elements of the set {r} according to relation(2) (applying the edge label value j=1 as determined in relation (5) forblock 1710) to provide an RPST, {3}, which is graphically illustrated inFIG. 24.

Execution of the initial instance of process 1800 then returns to block1812 for updating {RPSTs:: FT} by including {RPSTs:: ST} (={3} asdetermined above) and merged combinations of the enumerated {RPSTs:: ST}with any other previously enumerated RPSTs according to relation (4).Since {RPSTs:: FT} at this point comprises an empty set, block 1812merely updates {RPSTs:: FT} to include the single element of {RPSTs::ST}. Block 1814 updates the remainder as FT/ST=249/3=83. This numeralcorresponds to a subtree of tree 1900 formed by nodes 1902, 1906, 1908,1910 and 1912 graphically illustrated as subtree 2100 in FIG. 25.

On a second iteration of the processing loop of blocks 1806 through1816, block 1808 determines the non-composite factor of the remainderupdated at block 1814 of the first iteration of the processing loop.Here, the natural numeral remainder, 83 as determined at block 1814 inthe first iteration, comprises a non-composite numeral. Accordingly,block 1808 determines the natural numeral ST of the current iteration tobe 83. Block 1810 then determines {RPSTs:: 83} by initiating a thirdrecursive instance of process 1700. Since 83 is a non-composite naturalnumeral (as determined at diamond 1708), block 1710 determines theinverse push of 83 according to relation (3) as follows:

$\begin{matrix}\begin{matrix}{{{{push}^{- 1}\left( {{r = 1},{k = 2},{{ST} = 83}} \right)} = {< {child}}},{j >}} \\{{child} = {{Int}\left\lbrack {\left( {{P^{- 1}(83)} + 2 - \left( {2 - 1} \right)} \right)/2} \right\rbrack}} \\{= {{Int}\left\lbrack {\left( {23 + 2 - \left( {2 - 1} \right)} \right)/2} \right\rbrack}} \\{= 12} \\{j = \left\lbrack {\left( {{P^{- 1}(83)} + 2 - \left( {2 - 1} \right)} \right\rbrack{{modulo}\lbrack 2\rbrack}} \right.} \\{= \left\lbrack {\left( {23 + 2 - \left( {2 - 1} \right)} \right\rbrack{{modulo}\lbrack 2\rbrack}} \right.} \\{= 0}\end{matrix} & (6)\end{matrix}$

The result of this inverse push operation is graphically illustrated inFIG. 26 which includes a child tree 2200 of the subtree 2100 formed bythe nodes 1906, 1908, 1910 and 1912. Block 1712 then determines {RPSTs::12} by initiating a fourth recursive instance of process 1700 (settingFT=child=12). Since the natural numeral “12” is a composite numeral(representing a tree which merges subtrees at a root node), block 1718may determine {RPSTs:: 12} by initiating a first recursive instance ofprocess 1800. As block 1804 sets remainder=“12”, block 1808 determinesST (the natural numeral representing a first merged subtree) as “2.”Block 1810 may then determine {RPSTs:: 2} by initiating a fifthrecursive instance of process 1700. Here, since “2” is a non-compositenumeral (as determined at diamond 1708), block 1710 may determine theinverse push of “2” according to relation (3) as follows:

$\begin{matrix}\begin{matrix}{{{{push}^{- 1}\left( {{r = 1},{k = 2},{{ST} = 2}} \right)} = {< {child}}},{j >}} \\{{child} = {{Int}\left\lbrack {\left( {{P^{- 1}(2)} + 2 - \left( {2 - 1} \right)} \right)/2} \right\rbrack}} \\{= {{Int}\left\lbrack {\left( {1 + 2 - \left( {2 - 1} \right)} \right)/2} \right\rbrack}} \\{= 1} \\{j = \left\lbrack {\left( {{P^{- 1}(2)} + 2 - \left( {2 - 1} \right)} \right\rbrack{{modulo}\lbrack 2\rbrack}} \right.} \\{= \left\lbrack {\left( {1 + 2 - \left( {2 - 1} \right)} \right\rbrack{{modulo}\lbrack 2\rbrack}} \right.} \\{= 0}\end{matrix} & (7)\end{matrix}$Block 1712 may initiate a sixth recursive instance of process 1700 todetermine {RPSTs:: 1}. Diamond 1706 of the sixth recursive instance ofprocess 1700 may terminate and return {RPSTs:: 1}=Ø (i.e., the emptyset). Returning to the fifth recursive instance of process 1700, block1714 updates {RPSTs:: child} to include {r} ({r}={1} for this particularcase where tree 1200 is a BELT). Accordingly, {RPSTs:: 2}=push {1}={2}(using the edge label value j=0 as determined at block 1710 of the fifthrecursive instance of process 1700 and shown in relation (7)). Thiscorresponds with the RPST 2300 of child tree 2200 formed by node 1906,and either node 1908 or 1912 as shown in FIG. 27.

Returning to block 1812 of the first recursive instance of process 1800,{RPSTs:: 12} is updated as {2}. The remainder is updated to be thenatural numeral remainder/ST=12/2=6. Block 1808 determines ST to be thesmallest non-composite factor of the updated remainder (here, “6”) to be“2.” As illustrated above in the fifth recursive instance of process1700, block 1810 determines {RPSTs:: 2} to be {2} (again, correspondingwith the RPST of subtree 2200 formed by node 1906, and either node 1908or 1912). Block 1812 may then determine combinations of the previouslyenumerated elements of {RPSTs:: 12} with the elements of {RPSTs:: ST}using the set merger operation of relation (4) and update {RPSTs:: 12}as follows:

$\begin{matrix}\begin{matrix}{\left\{ {{RPSTs}{::}\mspace{14mu} 12} \right\} = {\left\{ {{RPSTs}{::}\mspace{14mu} 12} \right\} U\left\{ {{RPSTs}{::}\mspace{11mu} 2} \right\}{U\left\lbrack {\left\{ {{RPSTs}{::}\mspace{14mu} 12} \right\} \times} \right.}}} \\\left. \left\{ {{RPSTs}{::}\mspace{14mu} 2} \right\} \right\rbrack \\{= {\left\{ {{RPSTs}{::}\mspace{14mu} 2} \right\} U\left\{ {{RPSTs}{::}\mspace{14mu} 2} \right\}{U\left\lbrack {\left\{ {{RPSTs}{::}\mspace{14mu} 2} \right\} \times} \right.}}} \\\left. \left\{ {{RPSTs}{::}\mspace{14mu} 2} \right\} \right\rbrack \\{= {\left\{ 2 \right\} U\left\{ 2 \right\}{U\left\lbrack {\left\{ 2 \right\} \times \left\{ 2 \right\}} \right\rbrack}}} \\{= \left\{ {2,4} \right\}}\end{matrix} & (8)\end{matrix}$This updated {RPSTs:: 12} is graphically illustrated in FIG. 27(illustrating the RPST 2300 of child tree 2200) and 28 (illustrating anRPST 2400 of subtree 2200 associated with the natural numeral 4).

Block 1814 then updates the remainder=remainder/ST=6/2=3, and the nextiteration of the processing loop of blocks 1806 through 1816 determinesST as “3” at block 1808. Block 1810 may determine {RPSTs:: ST}={RPSTs::3}={3} as illustrated above in the first recursive instance of process1700. This resulting RSPT of the child tree 2200 includes nodes 1906 and1910 as shown in RPST 2500 of FIG. 29. Block 1812 then updates {RPSTs::12} (from {RPSTs:: 12}={2, 4} at relation (8)) as follows:

$\begin{matrix}\begin{matrix}{\left\{ {{RPSTs}{::}\mspace{14mu} 12} \right\} = {\left\{ {2,4} \right\} U\left\{ 3 \right\}{U\left\lbrack {\left\{ {2,4} \right\} \times \left\{ 3 \right\}} \right\rbrack}}} \\{= {{\left\{ {2,3,4} \right\} U\left\{ {6,12} \right\}} = \left\{ {2,3,4,6,12} \right\}}}\end{matrix} & (9)\end{matrix}$The resulting elements of {RPSTs:: 12} are graphically illustrated inFIG. 30 with corresponding nodes 1906, 1908, 1910 and 1912 of the childtree 2200 shown in FIG. 26. It should be noted that in the process ofenumerating of elements of the set {RPSTs:: 12} at relations (8) and(9), duplicate RPSTs were enumerated for the RPST corresponding withnumeral “4.” Here, in listing the elements of the set {RPSTs:: 12} atrelations (8) and (9) in this particular embodiment such a duplicatelisting the RPST corresponding with the numeral “4” was not included soas to provide unique, unordered elements of the set {RPSTs:: 12}.However, this is merely a particular embodiment provided forillustration and claimed subject matter is not limited in this respect.For example, it should be readily appreciated that the process ofenumerating RPSTs of a tree or subtree described herein may be readilyapplied alternative embodiments for enumerating RPSTs of a tree orsubtree that includes such duplicated RPSTs. Again, this example ofenumerating duplicate RPSTs is also merely an example provided for thepurpose of illustration and claimed subject matter is not limited inthis respect.

Returning to the third recursive instance of process 1700 (following theidentification of 2200 as the child tree of RPST 2100 at block 1710 andthe enumeration of the RPSTs of subtree 2200 as the elements of {RPSTs::12} in block 1712)), block 1714 updates {RPSTs:: child} to include{RPSTs:: 12} U {r}={1, 2, 3, 4, 6, 12}. Block 1716 may then complete theenumeration of the elements of {RPSTs:: 83} by performing a pushoperation on the elements of {RPSTs:: child} according to relation (2)(with label index value j=0 as determined in relation (6)) as follows:{RPSTs::83}=zero-push({1,2,3,4,6,12})={2,5,11,17,31,83}  (10)The resulting elements of {RPSTs:: 83} are graphically illustrated withreference to FIG. 31 with corresponding nodes 1902, 1906, 1908, 1910 and1912 of the subtree 2100 shown in FIG. 25.

Returning to the initial instance of process 1800 (following theenumeration of elements in {RPSTs:: 3} corresponding with a firstsubtree merged at root node 1902 as graphically illustrated in FIG. 24and the enumeration of elements in {RPSTs:: 83} corresponding with asecond subtree merged at root node 1902 as graphically illustrated inFIG. 31), block 1812 updates {RPSTs:: FT} as follows:

$\begin{matrix}\begin{matrix}{\left\{ {{RPSTs}{::}\mspace{14mu}{FT}} \right\} = {\left\{ {{RPSTs}{::}\mspace{14mu} 3} \right\} U\left\{ {{RPSTs}{::}\mspace{14mu} 83} \right\}{U\left\lbrack {\left\{ {{RPSTs}{::}\mspace{14mu} 3} \right\} \times} \right.}}} \\\left. \left\{ {{RPSTs}{::}\mspace{14mu} 83} \right\} \right\rbrack \\{= {\left\{ {2,3,5,11,17,31,83} \right\}{U\left\lbrack {\left\{ 3 \right\} \times \left\{ {2,5,11,17,31,} \right.} \right.}}} \\\left. \left. 83 \right\} \right\rbrack \\{= {\left\{ {2,3,5,11,17,31,83} \right\} U\left\{ {6,15,33,51,93,249} \right\}}} \\{= \left\{ {2,3,5,6,11,15,17,31,33,51,33,51,83,93,} \right.} \\\left. 249 \right\}\end{matrix} & (11)\end{matrix}$

While the above illustrated example is a specific case of enumeratingRPSTs from one particular BELT (associated with the natural numeral249), it should be understood that the processes are general enough toenumerate RPSTs for any tree. Also, while the illustrated example isspecifically directed to enumerating RPSTs of a BELT, claimed subjectmatter is not limited to this specific example or specifically to BELTs.

It should also be understood that, although particular embodiments havejust been described, claimed subject matter is not limited in scope to aparticular embodiment or implementation. For example, one embodiment maybe in hardware, such as implemented to operate on a device orcombination of devices, for example, whereas another embodiment may bein software. Likewise, an embodiment may be implemented in firmware, oras any combination of hardware, software, and/or firmware, for example.Such software and/or firmware may be expressed as machine-readableinstructions which are executable by a processor. Likewise, althoughclaimed subject matter is not limited in scope in this respect, oneembodiment may comprise one or more articles, such as a storage mediumor storage media. This storage media, such as one or more CD-ROMs and/ordisks, for example, may have stored thereon instructions, that whenexecuted by a system, such as a computer system, computing platform, orother system, for example, may result in an embodiment of a method inaccordance with claimed subject matter being executed, such as one ofthe embodiments previously described, for example. As one potentialexample, a computing platform may include one or more processing unitsor processors, one or more input/output devices, such as a display, akeyboard and/or a mouse, and/or one or more memories, such as staticrandom access memory, dynamic random access memory, flash memory, and/ora hard drive, although, again, claimed subject matter is not limited inscope to this example.

In the preceding description, various aspects of claimed subject matterhave been described. For purposes of explanation, specific numbers,systems and/or configurations were set forth to provide a thoroughunderstanding of claimed subject matter. However, it should be apparentto one skilled in the art having the benefit of this disclosure thatclaimed subject matter may be practiced without the specific details. Inother instances, well-known features were omitted and/or simplified soas not to obscure claimed subject matter. While certain features havebeen illustrated and/or described herein, many modifications,substitutions, changes and/or equivalents will now occur to thoseskilled in the art. It is, therefore, to be understood that the appendedclaims are intended to cover all such modifications and/or changes asfall within the true spirit of claimed subject matter.

1. A method comprising: identifying partial subtrees of a tree, saidtree comprising hierarchical data representing one or more XMLdocuments, said partial subtrees having depths up to a predetermineddepth defining a longest separation between a root node and terminalnode of said partial subtrees; enumerating rooted partial subtrees of atleast one of said identified partial subtrees; representing said tree asone or more electrical digital signals representing one or more targetnumerals, said one or more target numerals being associated with saidenumerated rooted partial subtrees according to an association betweentrees and numerals; determining one or more signals representing a probenumeral representing a query to said one or more XML documents based, atleast in part, on one or more electrical digital signals representing aprobe tree, said probe tree comprising a piece of information ofinterest; and comparing, using a processor, said one or more electricaldigital signals representing said probe numeral with said one or moresignals representing said one or more target numerals to detect apresence of said piece of information in said tree by detecting apresence of said one or more signals representing said probe tree in oneor more electrical digital signals representing said tree based, atleast in part, on a match of said one or more signals representing saidprobe numeral with one or more electrical digital signals representingat least one of said target numerals.
 2. The method of claim 1, whereinsaid probe numeral is based, at least in part, on said association oftrees and numerals.
 3. The method of claim 1, and further comprising:identifying non-terminal nodes of said tree as root nodes of saidpartial subtrees; and identifying one or more nodes descending from aroot node of at least one of said partial subtrees to be one or morenodes of said partial subtree.
 4. The method of claim 3, wherein saididentifying one or more nodes descending from a root node of at leastone of said partial subtrees further comprises identifying nodesdescending from a root node of at least one of said partial subtreesdown to said predetermined depth to be one or more nodes of said partialsubtree.
 5. The method of claim 3, wherein said identifying one or morenodes descending from a root node of at least one of said partialsubtrees further comprises identifying one or more nodes descending froma root node of at least one of said partial subtrees based, at least inpart, on electrical digital signals representing node and/or edge labelvalues associated with said nodes.
 6. The method of claim 1, and furthercomprising determining said one or more signals representing said probenumeral based, at least in part, on said association between trees andnumerals, and wherein said match corresponds with a partial subtree ofsaid tree matching said probe tree.
 7. The method of claim 1, whereinsaid tree comprises a binary edge labeled tree.
 8. The method of claim1, wherein said tree comprises hierarchical data representative of aknown biometric pattern and said probe numeral is representative of oneor more features of a subject and/or specimen.
 9. The method of claim 1,and further comprising further executing said instructions by saidprocessor to: determine target numerals associated with enumeratedrooted partial subtrees having a root node with at least two nodesdirectly descending nodes as composite natural numerals; and determinetarget numerals associated with enumerated rooted partial subtreeshaving a root node with a single directly descend nodes as non-compositenatural numerals.
 10. A method comprising: representing a tree as one ormore electrical digital signals representing one or more targetnumerals, said one or more target numerals being associated with partialsubtrees of said tree according to an association between trees andnumerals, said partial subtrees having depths up to a predetermineddepth defining a longest separation between a root node and terminalnode of said partial subtrees; determining one or more signalsrepresenting a probe numeral based, at least in part, on one or moreelectrical digital signals representing a probe tree, said probe treecomprising a piece of information of interest; and comparing, using aprocessor, said one or more signals representing said probe numeral withsaid one or more signals representing said one or more target numeralsto detect a presence of said piece of information in said tree bydetecting a presence of said one or more signals representing said probetree in one or more electrical digital signals representing said treebased, at least in part, on a match of said one or more signalsrepresenting said probe numeral with one or more electrical digitalsignals representing at least one of said target numerals, wherein saidtree comprises hierarchical data representative of a known biometricpattern and said probe numeral is representative of one or more featuresof a subject and/or specimen.
 11. An apparatus comprising: means,comprising one or more processors, for identifying subtrees of a tree,said tree comprising hierarchical data representing one or more XMLdocuments, said subtrees having depths up to a predetermined depthdefining a longest separation between a root node and terminal node ofsaid subtrees; means, comprising said one or more processors, forenumerating rooted partial subtrees of at least one of said identifiedsubtrees; means, comprising said one or more processors, forrepresenting said tree as one or more target numerals, said one or moretarget numerals being associated with said enumerated rooted partialsubtrees according to an association of trees and numerals; means,comprising said one or more processors, for determining a probe numeralrepresenting a query to said one or more XML documents based, at leastin part, on a probe tree, said probe tree comprising a piece ofinformation of interest; and means, comprising said one or moreprocessors, for comparing said probe numeral with said one or moretarget numerals to detect a presence of said piece of information insaid tree by detecting a presence of said probe tree in said tree based,at least in part, on a match of said probe numeral with at least one ofsaid target numerals.
 12. The apparatus of claim 11, wherein said probenumeral is based, at least in part, on said association of trees andnumerals.
 13. The apparatus of claim 11, wherein said means foridentifying said partial subtrees of said tree further comprises: meansfor identifying non-terminal nodes of said tree as root nodes of saidpartial subtrees; and means for identifying one or more nodes descendingfrom a root node of at least one of said partial subtrees to be one ormore nodes of said partial subtree.
 14. The apparatus of claim 13,wherein said means for identifying one or more nodes descending from aroot node of at least one of said partial subtrees further comprisesmeans for identifying one or more nodes descending from a root node ofat least one of said partial subtrees down to a predetermined depth tobe one or more nodes of said partial subtree.
 15. The apparatus of claim13, wherein said means for identifying one or more nodes descending froma root node of at least one of said partial subtrees further comprisesmeans for identifying nodes descending from a root node of at least oneof said partial subtrees based, at least in part, on node and/or edgelabel values associated with said nodes.
 16. The apparatus of claim 11,and further comprising means for determining said probe numeral based,at least in part, on said association between trees and numerals, andwherein said match corresponds with a partial subtree of said treematching said probe tree.
 17. The apparatus of claim 11, wherein saidtree comprises a binary edge labeled tree.
 18. The apparatus of claim11, and further comprising means for representing hierarchical datarepresentative of a known biometric pattern as said tree andrepresenting one or more features of a subject and/or specimen as saidprobe numeral.
 19. An apparatus comprising: means, comprising one ormore processors, for representing a tree as one or more target numerals,said one or more target numerals being associated with partial subtreesof said tree according to an association between trees and numerals,said partial subtrees having depths up to a predetermined depth defininga longest separation between a root node and terminal node of saidpartial subtrees; means, comprising said one or more processors, fordetermining a probe numeral based, at least in part, on one or moreelectrical digital signals representing a probe tree, said probe treecomprising a piece of information of interest; means, comprising one ormore processors, for comparing said probe numeral with said one or moretarget numerals to detect a presence of said piece of information insaid tree by detecting a presence of said probe tree in said tree based,at least in part, on a match of said probe numeral with at least one ofsaid target numerals; and means, comprising one or more processors, forrepresenting hierarchical data representative of a known biometricpattern as said tree and representing one or more features of a subjectand/or specimen as said probe numeral.
 20. An article comprising: astorage medium comprising machine-readable instructions stored thereonwhich are executable by a processor to: identify subtrees of a tree,said tree comprising hierarchical data representing one or more XMLdocuments, said subtrees having depths up to a predetermined depthdefining a longest separation between a root node and terminal node ofsaid subtrees; enumerate rooted partial subtrees of at least one of saididentified subtrees; represent said tree as one or more target numerals,said one or more target numerals being associated with said enumeratedrooted partial subtrees according to an association between trees andnumerals; determine a probe numeral representative of a query to saidone or more XML documents based, at least in part, on a probe tree, saidprobe tree comprising a piece of information of interest; and comparesaid probe numeral with said one or more target numerals to detect apresence of said piece of information in said tree by detecting apresence of said probe tree in said tree based, at least in part, on amatch of said probe numeral with at least one of said target numerals.21. The article of claim 20, wherein said machine-readable instructionsare further executable by said processor to generate said one or moretarget numerals based, at least in part, on one or more numeralsrepresenting rooted partial subtrees of at least one of said partialsubtrees.
 22. The article of claim 20, wherein said machine-readableinstructions are further executable by said processor to determine saidprobe numeral based, at least in part, on said association between treesand numerals.
 23. The article of claim 20, wherein said instructions arefurther executable by said processor to: identify non-terminal nodes ofsaid tree as root nodes of said partial subtrees; and identify one ormore nodes descending from a root node of at least one of said partialsubtrees to be one or more nodes of said partial subtree.
 24. Thearticle of claim 23, wherein said instructions are further executable bysaid processor to identify one or more nodes descending from a root nodeof at least one of said partial subtrees down to said predetermineddepth to be one or more nodes of said partial subtree.
 25. The articleof claim 23, wherein said instructions are further executable by saidprocessor to identify one or more nodes descending from a root node ofat least one of said partial subtrees to be one or more nodes of saidpartial subtree based, at least in part, on node and/or edge labelvalues associated with said one or more nodes.
 26. The article of claim20, wherein said instructions are further executable by said processorto determine said probe numeral based, at least in part, on saidassociation between trees and numerals, and wherein said matchcorresponds with a partial subtree of said tree matching said probetree.
 27. The article of claim 20, wherein said tree comprises a binaryedge labeled tree.
 28. The article of claim 20, wherein saidinstructions are further executable by said processor to representhierarchical data representative of a known biometric pattern as saidtree and represent one or more features of a subject and/or specimen assaid probe numeral.
 29. An article comprising: a storage mediumcomprising machine-readable instructions stored thereon which areexecutable by a processor to: represent a tree as one or more targetnumerals, said one or more target numerals being associated withidentified partial subtrees of said tree according to an associationbetween trees and numerals, said identified partial subtrees havingdepths up to a predetermined depth defining a longest separation betweena root node and a terminal node of said partial subtrees; determine aprobe numeral based, at least in part, on a probe tree, said probe treecomprising a piece of information of interest; compare said probenumeral with said one or more target numerals to detect a presence ofsaid piece of information in said tree by detecting a presence of saidprobe tree in said tree based, at least in part, on a match of saidprobe numeral with at least one of said target numerals; and representhierarchical data representative of a known biometric pattern as saidtree and represent one or more features of a subject and/or specimen assaid probe numeral.
 30. An apparatus comprising: a computing platformcomprising one or more processors programmed with instructions to:identify subtrees of a tree, said tree comprising hierarchical datarepresenting one or more XML documents, said subtrees having depths upto a predetermined depth defining a longest separation between a rootnode and a terminal node of said subtrees; enumerate rooted partialsubtrees of at least one of said identified subtrees; represent saidtree as one or more electrical digital signals representing one or moretarget numerals, said one or more target numerals being associated withsaid enumerated rooted partial subtrees according to an associationbetween trees and numerals; determine a probe numeral representative ofa query to said one or more XML documents based, at least in part, saidprobe tree comprising a piece of information of interest; and compareone or more electrical digital signals representing said probe numeralwith said one or more signals representing said one or more targetnumerals to detect a presence of said piece of information in said treeby detecting a presence of said probe tree in said tree based, at leastin part, on a match of said probe numeral with at least one of saidtarget numerals.
 31. The apparatus of claim 30, wherein said one or moreprocessors are further programmed with instructions to determine saidone or more signals representing said probe numeral based, at least inpart, on an association of trees and numerals.
 32. The apparatus ofclaim 30, wherein said one or more processors are further programmedwith instructions to: identify said one or more partial subtrees of saidtree; and determine said one or more signals representing said one ormore target numerals based, at least in part, on at least a portion ofsaid partial subtrees and according to said association between treesand numerals.
 33. The apparatus of claim 30, wherein said one or moreprocessors are further programmed with instructions to: identifynon-terminal nodes of said tree as root nodes of said partial subtrees;and identify one or more nodes descending from a root node of at leastone of said partial subtrees to be one or more nodes of said partialsubtree.
 34. The apparatus of claim 33, wherein said one or moreprocessors are further programmed with instructions to identify one ormore nodes descending from a root node of at least one of said partialsubtrees down to said predetermined depth to be one or more nodes ofsaid partial subtree.
 35. The apparatus of claim 33, wherein said one ormore processors are further programmed with instructions to identify oneor more nodes descending from a root node of at least one of saidpartial subtrees to be one or more nodes of said partial subtree based,at least in part, on one or more electrical digital signals representingnode and/or edge label values associated with said one or more nodes.36. The apparatus of claim 30, wherein said one or more processors arefurther adapted to determine said one or more signals representing saidprobe numeral based, at least in part, on said association between treesand numerals, and wherein said match corresponds with a partial subtreeof said tree matching said probe tree.
 37. The apparatus of claim 30,wherein said tree comprises a binary edge labeled tree.
 38. Theapparatus of claim 30, wherein said one or more processors are furtherprogrammed with instructions to represent hierarchical datarepresentative of a known biometric pattern as said tree and representone or more features of a subject and/or specimen as said one or moresignals representing said probe numeral.
 39. An apparatus comprising: acomputing platform comprising one or more processors programmed withinstructions to: represent a tree as one or more electrical digitalsignals representing one or more target numerals, said one or moretarget numerals being associated with identified partial subtrees ofsaid tree according to an association between trees and numerals, saidpartial subtrees having depths up to a predetermined depth defining alongest separation between a root node and a terminal node of saidpartial subtrees; determine one or more signals representing a probenumeral based, at least in part, on one or more electrical digitalsignals representing a probe tree, said probe tree comprising a piece ofinformation of interest; compare one or more electrical digital signalsrepresenting said probe numeral with said one or more signalsrepresenting said one or more target numerals to detect a presence ofsaid piece of information in said tree by detecting a presence of saidone or more signals representing said probe tree in one or moreelectrical digital signals representing said tree based, at least inpart, on a match of said one or more signals representing said probenumeral with one or more electrical digital signals representing atleast one of said target numerals; and represent hierarchical datarepresentative of a known biometric pattern as said tree and representone or more features of a subject and/or specimen as said one or moresignals representing said probe numeral.